Examples Of Hopf Algebras Research Articles

Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations

Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.

On Indecomposable Ideals Over Some Algebras

In this paper we investigate a family of algebras endowed with a suitable non-degenerate bilinear form that can be used to define two different notions of dual for a given right ideal. We apply our results to the classification of the right ideals and their duals in the cyclic group algebra, in the Taft algebra and in another example of Hopf algebra arising as bosonization.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.

Orderings and *-Orderings on Cocommutative Hopf Algebras

Our aim is to construct new examples of totally ordered and ∗-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings \(\Bbbk\)G and smash products U(L)\(\# _{\varphi } \Bbbk\)G. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of ∗-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for \(\Bbbk = \mathbb{R}\) and L finite–dimensional, we characterize the orderability of U(L)\(\# _{\varphi } \Bbbk\)G; for \(\Bbbk = \mathbb{C}\), we give a necessary and a sufficient condition for ∗-orderability of \(\Bbbk\)G (G orderable, respectively, G residually ‘torsion-free nilpotent’). Moreover, for \(\Bbbk = \mathbb{C}\) and L finite-dimensional, we reduce the problem of characterizing the ∗-orderability of U(L)\(\# _{\varphi } \Bbbk\)G to the problem of characterizing the ∗-orderability of \(\Bbbk\)G. The latter remains open.

A Generalization of a Two-Parameter Quantization for the Group GL2(k)

We generalize a well-known two-parameter quantization for the group GL2(k) (over an arbitrary field k). Specifically, a certain class of Hopf algebras is constructed containing that quantization. The algebras are constructed given an arbitrary coalgebra and an arbitrary pair of its commuting anti-isomorphisms, and are defined by quadratic relations. They are densely linked to the compact quantum groups introduced by Woronowicz. We give examples of Hopf algebras that can be rowed up to the two-parameter quantization for GL2(k).

On identities for coalgebras and hopf algebras

The notion of a polynomial identity for algebras is well-known. In this work the notion of identity is transferred to coalgebras (where one may call it a coidentity). The author studies the basic properties of coidenti-ties, some examples of Hopf algebras with an identity or a coidentity are considered.

On the nonlinearity interpretation ofq- andf-deformation and some applications

q-oscillators are associated to the simplest non-commutative example of Hopf algebra and may be considered to be the basic building blocks for the symmetry algebras of completely integrable theories. They may also be interpreted as a special type of spectral nonlinearity, which may be generalized to a wider class of f-oscillator algebras.

Lots of Hopf Algebras

The purpose of this paper is to give a “universal” construction of Hopf algebras over any commutative ring. The mod-pSteenrod algebras are all examples of this construction; other examples of Hopf algebras related to the mod-pSteenrod algebras are also produced. Then we give an argument to produce deformations of these Hopf algebras via “twisting.” The techniques of this paper are, in a way, “scheme-theoretic,” producing many rather complicated Hopf algebras, only a few of which are studied in any detail here; however, we hope to return to the study of some of these Hopf algebras in future work.

K 0-rings and twisting of finite dimensional semisimple hopf algebras

We show that the K 0-group of a finite dimensional semisimple Hopf algebra has a natural structure of a ring with involution and prove that this ring is a twisting invariant of the Hopf algebra.We also study relations between algebraic structure of a Hopf algebra and the one of its K 0ring and prove that the twisting of a finite simple group is a simple Hopf algebra (i.e., it does not have proper normal Hopf subalgebras).In this case the dual Hopf algebra does not have proper Hopf subalgebras.Further, we construct several series of non trivial Hopf algebras by the twisting of the classical series of finite groups and give examples of Hopf algebras which can not be described as twisting of any group.